Chemin, Gallagher, and Paicu obtained in 2010 a class of large initial data that generate a global smooth solution to the three-dimensional, incompressible Navier–Stokes equation. The data varies slowly in the vertical direction — it is expressed as a function of — and it has a norm that blows up as the small parameter goes to zero. This type of initial data can be regarded as an ill prepared case, in contrast with the well prepared case treated in earlier papers. The data was supposed to evolve in a special domain, namely . The choice of a periodic domain in the horizontal variable played an important role.
The aim of this article is to study the case where the fluid evolves in the whole space . In this case, we have to overcome the difficulties coming from very low horizontal frequencies. We consider in this paper an intermediate situation between the well prepared case and ill prepared situation (the norms of the horizontal components of initial data are small but the norm of the vertical component blows up as the small parameter goes to zero). The proof uses the analytical-type estimates and the special structure of the nonlinear term of the equation.
"Global regularity for the Navier–Stokes equations with some classes of large initial data." Anal. PDE 4 (1) 95 - 113, 2011. https://doi.org/10.2140/apde.2011.4.95